MINIMIZATION OF THE LOWEST EIGENVALUE FOR A VIBRATING BEAM

. In this paper we solve the minimization problem of the lowest eigenvalue for a vibrating beam. Firstly, based on the variational method, we establish the basic theory of the lowest eigenvalue for the fourth order measure diﬀerential equation (MDE). Secondly, we build the relationship between the minimization problem of the lowest eigenvalue for the ODE and the one for the MDE. Finally, with the help of this built relationship, we ﬁnd the explicit optimal bound of the lowest eigenvalue for a vibrating beam.


1.
Introduction. The vibrating beam investigated in this paper is subject to an axial compressive load λ which causes it to buckle. The beam is supported on an elastic foundation which provides, at each point x, an elastic destructive force F (x)y, which opposes restoration toward the line of no deflection and is directly proportional to the displacement y. From elementary beam theory, the natural modes of buckling of our problem are the eigenfunctions of a vibrating beam descried by y (t) + λy (t) + F (t)y(t) = 0, t ∈ [0, 1], (1) subject to elastically constrained boundary conditions. Here, we will assume hingedhinged boundary conditions y(0) = y (0) = 0 = y(1) = y (1). ( See [10]. The lowest eigenvalue λ 1 represents the smallest axial compressive force necessary to cause the beam to buckle. In this paper we are concerned with the lowest eigenvalue λ 1 (F ) and will give its optimal lower bound when the L 1 norm F 1 = F L 1 ([0,1]) is bounded. To this end, we will solve the following minimization problem L(r) := inf{λ 1 (F ) : Here, for r ∈ (0, +∞), is the ball of (L 1 , · 1 ). Once minimization problem (3) is solved, one has the following lower bound for λ 1 (F ): which will be shown to be optimal in a certain sense.
Since the L 1 balls B 1 [r] lack compactness even in the weak topology of L 1 , we usually do not know whether minimization problem (3) can be attained by some potentials from B 1 [r]. To overcome this, different from the approach in [7,14,15,16,17], we here will extend the problem to the measure case.
Firstly, based on the variational method, we will establish the minimization characterization for the lowest eigenvalue λ 1 (µ) of the fourth order measure differential equation (MDE) described by dy (3) (t) + λy (t) dt + y(t) dµ(t) = 0, t ∈ [0, 1], with the corresponding boundary condition (2), arriving at the following first contribution of this paper.
Theorem 1.1. Let µ ∈ M 0 and the Rayleigh form Problem (5)-(2) admits a lowest eigenvalue λ 1 (µ), which has the following minimization characterizations Here, M 0 , W 2,2 0 and H 3 00 are as in (9),(25) and (26), respectively. Secondly, we build a relation between the minimization problem of the lowest eigenvalue for the ODE (1) and the one for the MDE (5) and use this relationship to find the explicit optimal lower bound of the lowest eigenvalue for the ODE (1), arriving at the following main contribution of this paper. Here, the invertible elementary function Q : (−∞, π 2 ) → (0, +∞) is defined as

MINIMIZATION OF THE LOWEST EIGENVALUE 2081
This paper is organized as follows. In Section 2, we will recall basic facts on measures, the Lebesgue-Stieltjes integral and the Riemann-Stieltjes integral. In Section 3, we will use the variational method to prove Theorem 1.1. In section 4, after obtaining the relationship between the minimization problem for ODE and the one for MDE, described by Theorem 4.1, we will find the explicit optimal bound of the lowest eigenvalue for the vibrating beam and thus prove Theorem 1.2.
2. Basic facts on measures. Let I = [0, 1]. For a function µ : I → R, the total variation of µ (over I) is defined as be the space of non-normalized R-valued measures of I. Here, for any t ∈ [0, 1), µ(t+) := lim s↓t µ(s) is the right-limit. The space of (normalized) R-valued measures is For simplicity, we write V(µ, I) as µ V . By the Riesz representation theorem [5], (M 0 (I, R), · V ) is the same as the dual space of the Banach space (C(I, R), · ∞ ) of continuous R-valued functions of I with the supremum norm · ∞ . In fact, which refers to the Riemann-Stieltjes integral [1]. Moreover, one has Thenν ∈ M 0 (I, R) satisfies ν V =ν(1) −ν(0) and for any f ∈ C(I, R) and Typical examples of measures are as follows.
• Let : I → R be (t) ≡ t. Then yields the Lebesgue measure of I and the Lebesgue integral. More generally, any q ∈ L 1 (I, R) induces an absolutely continuous measure defined by In this case, one has and QUANYI LIANG, KAIRONG LIU, GANG MENG AND ZHIKUN SHE for any f ∈ C(I, R) and subinterval I 0 ⊂ I.
• For a = 0, the unit Dirac measure at t = 0 is • For a ∈ (0, 1], the unit Dirac measure at t = a is In the space M 0 (I, R) of measures, besides the usual topology induced by the norm · V , one has the following weak * topology w * . Definition 2.2. Let µ 0 , µ n ∈ M 0 (I, R), n ∈ N. We say that µ n is weakly * convergent to µ 0 if and only if one has In general, a measure cannot be a limit of smooth measures in the norm · V . However, in the w * topology, the following conclusion holds.
Moreover, if µ 0 is increasing (decreasing) on I, then the sequence {µ n } above can be chosen such that for each n ∈ N, µ n is increasing (deceasing) on I and µ n V = µ 0 V .
Considering q ∈ L 1 (I, R) as a density, one has the measure or distribution given by (13). Since µ q V = q 1 , Via (13), by the Hölder inequality and the isometrical embedding (17), one has the following result on the L 1 balls B 1 [r] and the M 0 balls B 0 [r] [8].
The following inclusion is proper As for the compactness of these balls in weak * topology, we have the following result.
3. Characterization for the lowest eigenvalue of MDE. Firstly, given a measure µ ∈ M 0 := M 0 (I, R), let us consider the fourth order linear MDE with the measure µ, formulated as: The initial condition of the MDE formulated by (19) can be written as Since we have assumed that y ∈ C, the right-hand sides of (20), (21), (22) are the Lebesgue integral and the right-hand side of (23) is the Lebesgue-Stieltjes integral, respectively.
One can find a proof from [3,13] based on the Kurzweil-Stieltjes integral to show the existence and uniqueness of solution for (19)-(24).
Here denotes the derivative with respect to t, is the space of continuously differentiable functions and AC([0, 1], R) is the space of absolutely continuous functions.
We use y(t, y 0 , y 1 , y 2 , y 3 ) to denote the unique solution of (19)-(24). Let be the fundamental solutions of (19). By the linearity of (19) and the uniqueness of solution, one has that, for t ∈ [0, 1], For the second order linear MDE, the continuity of solutions in measures has been obtained in [8]. In a similar way, for the fourth order linear MDE, we can also easily prove the following conclusion.
Further, by Corollary 1, we have the following corollary.
Secondly, we consider eigenvalue problem of the fourth order equation (5) with the boundary condition (2). It is a standard result that any (possible) eigenvalue λ ∈ R of problem (5)-(2) with eigenfunction u ∈ H 3 00 must satisfy where R(·) is as in (6). We will show that problem (5)-(2) does admit the lowest eigenvalue. To this end, we need the following basic estimate.
Proof. Assuming u ∈ H 2 0 \{0}, we have By (34) and (35), we have that Based on Lemma 3.4 and the variational method, we can now use R(u) to prove Theorem 1.1 for the minimization characterizations of the lowest eigenvalue as follows.
Then, by (37), Combining with the assumption that u n ∞ = 1, it is easy to see that {u n } ⊂ H 2 0 is bounded. Since H 2 0 is a Hilbert space and thus is compactly embedded into C 1 , there exists a non-zero u 0 ∈ H 2 0 such that u n → u 0 in (H 2 0 , w) and u n → u 0 in (C 1 , · C 1 ), going to a subsequence if necessary. Thus  (u n ) 2 dt.
Corollary 3. Let µ 1 , µ 2 ∈ M 0 . Then Additionally, the continuity of the lowest eigenvalue in measures with the weak * topology can be proved by the same arguments as those in [4]. Theorem 3.5. As a nonlinear functional, λ 1 (µ) is continuous in µ ∈ (M 0 , w * ).
4. Minimization of the lowest eigenvalue for a vibrating beam. In this section, for minimizing the lowest eigenvalue for the vibrating beam, we will firstly build the relationship between minimization problem of the lowest eigenvalue for the ODE (1) and the one for the MDE (5) as follows.
Theorem 4.1. Given r > 0, consider the following minimization problem One has L(r) =L(r).
On the other hand, there isμ ∈ B 0 [r] such that λ 1 (μ) =L(r) because B 0 [r] is sequentially compact in (M 0 , w * ) and λ 1 (µ) is continuous in µ ∈ (M 0 , w * ) by Theorem 3.5. According to the property of measures in Lemma 2.1 and the monotonicity of λ 1 (µ) in Corollary 3, without loss of generality, we can assume that µ = −μ is decreasing. Then, by Lemma 2.3, there exists a sequence of measures Therefore, by Theorem 3.5, we havẽ From (45) and (46), we have that L(r) =L(r). Now, based on the relationship (44) between minimization problem of the ODE and the MDE described by Theorem 4.1, we can then useL(r) to find the explicit optimal lower bound of the lowest eigenvalue for the vibrating beam (1) and thus prove Theorem 1.2 as follows.
The proof of Theorem 1.2. We will solve this extremal value problem in three steps.